The connection between mathematics and art goes back thousands of years. Mathematics has been used in the design of Gothic cathedrals, Rose windows, oriental rugs, mosaics and tilings. Geometric forms were fundamental to the cubists and many abstract expressionists, and award-winning sculptors have used topology as the basis for their pieces. Dutch artist M.C. Escher represented infinity, Möbius bands, tessellations, deformations, reflections, Platonic solids, spirals, symmetry, and the hyperbolic plane in his works.

Mathematicians and artists continue to create stunning works in all media and to explore the visualization of mathematics--origami, computer-generated landscapes, tesselations, fractals, anamorphic art, and more.

 "Dissection Dominoes," by Margaret Kepner (Washington, DC)50 x 50 cm, archival inkjet print, 2016 This piece contains 24 'dominoes' and one square, arranged in a spiral pattern. Each domino has two shapes: a white square and a black polygram (or star). These shapes are cut into subparts so as to be geometric dissections of each other. For example, the upper-right domino displays a dissection of a square into seven pieces that can be reassembled into the adjacent decagon. The domino directly below also displays a dissection of a square into a polygram with 10 vertices, but with every second one connected ({10/2}). Continuing down the right-hand side, there are dominoes showing dissections of the {10/3} and {10/4} shapes. This work provides opportunities for studying the properties--and enjoying the beauty--of geometric dissections. --- Margaret Kepner May 09, 2017
 "The Fish," by Umut Isik (University of California, Irvine, CA)28 x 43 cm, print, 2016 I have built a software infrastructure for producing artworks from simple mathematical functions. This creates a new medium for artistic and mathematical expression; one where it more natural to work with simple mathematical descriptions rather than imperative/iterative processes. In this work these algebraic curves divide the plane into many regions. I colored these regions using a probability distribution that produces a mixture of strong and light colors. --- Umut Isik May 09, 2017
 "Moore's Parterre," by Lana Holden (Skew Loose, LLC, Terre Haute, IN)60 x 60 cm, cotton, wool, polyamide, and silk yarn, 2014 The essence of Bruges crochet technique is the making of a crocheted tape that is shaped as it is made into a curve to form dense lace. I became interested in exploring it for creating space-filling curves. I experimented with the classic Hilbert curve for early studies, but chose Moore's variation for the larger work for a couple of reasons. First, the inherent symmetries of Moore curves are aesthetically pleasing. More importantly, in Moore curves, the starting point and endpoint are adjacent points, allowing the piece to be a closed loop. (Can you find the join?) The long color sections of the yarn used display the point clustering properties of Hilbert/Moore curves. --- Lana Holden May 09, 2017
 "Champy," by George Hart (Stony Brook University, Stony Brook, NY)32 x 32 x 32 cm, laser-cut wood, stained, 2016 Thirty components suggestive of "sea monsters" dance around each other, only touching at the hands and mouths. The arrangement of the thirty identical planar parts comes from the face planes of a rhombic triacontahedron, which provides a mathematical foundation for the structure. There are six parts in each of five colors, arranged with a five-color pattern based on the compound of five cubes. The order of the five colors of heads is different around each five-sided opening---all the even permutations. This was a prototype model for a larger (4-foot diameter) version of this design, installed at the Burr and Burton Academy in Vermont. The name "Champy" comes from the legend of a reputed lake monster said to be living in Lake Champlain. --- George Hart May 09, 2017
 "The Double-Knitting Groups," by Susan Goldstine (St. Mary's College of Maryland, St. Mary's City, MD)40 x 40 cm, silk/merino yarn, 2016 The Double-Knitting Groups exhibits all of the wallpaper symmetry types possible in double knitting. For a harmonious overall design, I grouped the nine structures into three pattern families: scrolls, hearts, and vines. Roughly speaking, the symmetry groups get more complex from top left to bottom right: the upper left pattern has only translations, the three adjacent to it have only translations and one other type of plane symmetry (clockwise: reflections, rotations, and glide reflections), and the remaining patterns have at least three symmetry types. --- Susan Goldstine May 09, 2017
 "Hexa Form," by Robert Fathauer (Tessellations Company, Phoenix, AZ)20 x 20 x 20 cm, ceramics, 2016 I'm endlessly fascinated by certain aspects of our world, including symmetry, chaos, and infinity. Mathematics allows me to explore these topics in distinctive artworks that I feel are an intriguing blend of complexity and beauty. This abstract sculpture, which mixes biological and geometric forms, is based on a regular hexahedron (more commonly known as a cube). It retains most, but not all, of the symmetries of the cube. --- Robert Fathauer May 09, 2017
 "They Arrive," by Frank A. Farris (Santa Clara University San Jose, CA)20 x 25 cm, digital aluminum print on DuraPlaq mount, 2016 Glowing globes with three types of polyhedral symmetry drift over a moonlit mountain to land on the lake. Are they wafting from the Platonic world into ours? (The patterns on the globe were created with domain colorings of meromorphic functions invariant under the actions of the three chiral polyhedral groups. In the past, I always used rectangular photographs to paint spheres, resulting in images with singularities. New techniques allow the source photograph to live on the Riemann sphere, allowing poles to be painted just as if they were zeroes. Ray tracing and manipulation of the original daytime mountain photograph were done in Photoshop.) --- Frank A. Farris May 09, 2017
 "Fractal Monarchs," by Doug Dunham and John Shier (University of Minnesota, Duluth)Best photograph, painting, or print - 2017 Mathematical Art Exhibition 30 x 40 cm, color printer, 2016 This is a fractal pattern whose motifs are monarch butterflies. We modify our usual rule that motifs cannot overlap by allowing the antennas - but not the rest of the motif - to overlap another motif. Expanding on the area rule of the Goals statement, the area of the n-th motif is given by A/(zeta(c,N)(N+n)^c), where A is the area of the region, and zeta(c,N) is the Hurwitz zeta function, a generalization of the Riemann zeta function (for which N = 1; our algorithm starts with n = 0). For this pattern c = 1.26, N = 1.5, and 150 butterflies fill 72% of the bounding rectangle. --- Doug Dunham May 09, 2017
 "Klein Bottle," by Jennifer Doyle (Lawrence Berkeley National Laboratory Berkeley, CA)23 x 23 x 20 cm, galvanized steel wire, 2015 My first experiments with mathematically-themed wire sculpture have been Klein bottles. The concept of producing a sculptural Klein bottle fascinates me, as the two concepts seem to be at odds: a sculpture is, by its nature, 3-dimensional, yet a Klein bottle is not; a sculpture, by its nature, has volume, yet a Klein bottle does not. The "classic" Klein bottle is considerably more "bottle-shaped" than my piece; I decided to shape my piece into a form more resembling two semi-toroids. --- Jennifer Doyle May 09, 2017
 "Secret Hexagons," by Moira Chas (Stony Brook University, Stony Brook, NY)5 x 35 x 35 cm, crochet This piece addresses the question: What is the maximum number of regions a surface can be divided into, so that each pair of regions share a length of their border (vertices don’t count)? In the torus, the maximum number is seven. --- Moira ChasMay 09, 2017
 "Map of Invariability I," by Conan Chadbourne (San Antonio, TX)60 x 60 cm, archival digital print, 2016 My work is motivated by a fascination with the occurrence of mathematical and scientific imagery in traditional art forms, and the mystical, spiritual, or cosmological significance that is often attached to such imagery. The Klein Quartic is a genus-three Riemann surface which can be covered by a regular tessellation of 24 heptagons. In this image, the Klein quartic is projected into the Poincaré disk, and this heptagonal tessellation is given a regular 8-coloring. Each triplet of heptagons of any given color is fixed by a subgroup of order 21 of the full automorphism group of the surface. --- Conan Chadbourne May 09, 2017
 "Streamlines," by Anne Burns (professor emerita, Long Island University, Huntington, NY)48 x 32 cm, digital print, 2016 I began my studies as an art major, but switched to mathematics. When I went to my first conference on fractals I was hooked. Visualizing mathematical concepts allowed me to combine both of my interests. The streamlines of the vector field dx/dt = x^2 - y^2, dy/dt = 2xy (the real and imaginary parts of the complex function f(z) = z^2 ) are the directed paths along which the tangent vector is equal to (dx/dt, dy/dt). They are circles tangent to the real axis. the attached vectors are colored according to their slope. --- Anne Burns May 09, 2017
 "Penrose Skates," by Douglas Burkholder (Lenoir-Rhyne University, Hickory, NC)50 x 50 cm, digital print, 2016 This artwork evolved from a search for beauty and patterns within Penrose’s non-periodic tiling of the plane with kites and darts. Half darts and half kites can be repeatedly subdivided into five smaller components. Start by labeling these five subcomponents A-E. Then, similar to creating the Sierpinski triangle, alternately subdivide and remove all the components with a certain label. After removing tile type A for three iterations we change to removing tile type B for five more iterations. Instead of painting the tiles remaining, pursuant curves are constructed on the regions removed. --- Douglas Burkholder May 09, 2017
 "Bicycle Wreck," by Fielding Brown (Westwood, MA)45 x 45 x 45 cm, wood & aluminum frame, silvered mylar strings, 2016 My sculptures are Lissajous figures in 3D. This sculpture is in two parts: an outer frame of laminated hardwood and aluminum, and an inner web of strings. The frame is defined by three, simultaneous, parametric equations, time as parameter. The strings are silvered, package-wrap string, space-defined by a commercial graphing calculator program. The web of strings guides the eye to see imaginary lines and surfaces. --- Fielding Brown May 09, 2017
 "Twelve Pi," by Sarah Berube (Diametric Arts, Shutesbury, MA)5 x 5 x 5 cm, 3D printed steel, 2016 My work currently revolves around the polyhedral symmetry groups. I pursue this form of art because I am strongly drawn to symmetry and the satisfying sense of beauty and perfection it evokes. My design process is an exploration of the properties of polyhedra. The design for this sculpture is comprised of twelve pi symbols arranged with tetrahedral symmetry. At four different locations, the right legs of three pi symbols spiral into each other, forming vortices in lieu of vertices. --- Sarah Berube May 09, 2017
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 Art & Music, MathArchives Geometry in Art & Architecture, by Paul Calter (Dartmouth College) Harmony and Proportion, by John Boyd-Brent International Society of the Arts, Mathematics and Architecture Journal of Mathematics and the Arts Mathematics and Art, the April 2003 Feature Column by Joe Malkevitch Maths and Art: the whistlestop tour, by Lewis Dartnell Mathematics and Art, (The theme for Mathematics Awareness Month in 2003) MoSAIC - Mathematics of Science, Art, Industry, Culture Viewpoints: Mathematics and Art, by Annalisa Crannell (Franklin & Marshall College) and Marc Frantz (Indiana University) Visual Insight, blog by John Baez